On conjugacy of second-order half-linear differential equations on the real axis

Some conjugacy criteria are given for the equation $$\big(|u'|^{\alpha}\operatorname{sgn}u'\big)'+p(t)|u|^{\alpha}\operatorname{sgn} u=0,$$ where $p\colon\mathbb{R} \to \mathbb{R}$ is a locally integrable function and $\alpha>0$, which generalise and supplement results known in the existing literature. Illustrative examples justifying applicability of the main results are given, as well. The results obtained are new even for linear differential equations, i.e., if $\alpha=1$

Existence and exact multiplicity of positive periodic solutions to forced non-autonomous Duffing type differential equations

The paper studies the existence, exact multiplicity, and a structure of the set of positive solutions to the periodic problem u''=p(t)u+q(t,u)u+f (t); u(0)=u(\omega), u'(0)=u'(\omega), where p, f\in L([0,\omega]) and q : [0,\omega]\times R\to R is Carathéodory function. Obtained general results are applied to the forced non-autonomous Duffing equation u'' = p(t)u+h(t)|u|^\lambda\sgn u+f (t), with \lambda>1 and a non-negative h\in L([0,\omega]). We allow the coefficient p and the forcing term f to change their signs

Solvability of a periodic type boundary value problem for first order scalar functional differential equations

summary:Nonimprovable sufficient conditions for the solvability and unique solvability of the problem $$u^{\prime }(t)=F(u)(t)\,,\qquad u(a)-\lambda u(b)=h(u)$$ are established, where $F:\rightarrow$ is a continuous operator satisfying the Carathèodory conditions, $h:\rightarrow R$ is a continuous functional, and $\lambda \in$